Hamiltonian of an optimal-control problem ($H(t,x,u,p)=L(t,x,u)+p\cdot f(t,x,u)$)

In a normal minimization convention for optimal control, the running cost $L$ and dynamics $\dot{x}=f$ define $H=L+p\cdot f$. The multiplier $p$ is a costate; this Hamiltonian is a device for variational necessary conditions rather than necessarily a physical energy.